We study a class of analytic
functions which unifies a number of classes previously studied, including functions
with boundary rotation at most kπ, functions convex of order ρ and the Robertson
functions, i.e., functions f for which zf′ is α-spirallike. We obtain representation
theorems for this general class, and using a simple variational formula, also obtain
sharp bounds on the modulus of the second coefficient of the series expansion of these
functions. Using a univalence criterion due to Ahlfors, we determine a condition on
the parameters k, α, and ρ which will ensure that a function in this class is univalent.
This result improves previously published results for various subclasses and
is sharp for the class of functions f for which zf′ is α-spirallike of order
ρ.
